Explain how the bank works, where your deposits actually go, what the bank does with their savings and that it is safe, secure and that they are earning money on their money!
Younger the better and continue to take them to the bank to physically deposit their savings. This reinforces a strong message about saving.
I understand that depositing at the ATM or via the iPhone is a lot more convenient but I don't think they get the same message that way.
I am not referring to your kids college savings plan or 529 plan, that is something that should be handled differently.
Pre-school and kindergarten teachers use money, usually pictures of coins to teach our kids how to count and add. Your job is simply to take it one step further.
Explain that this many coins is what it costs to buy their favorite toy or ice cream.
This relationship is important to learn early. Observation and repetition are two important ways that kids learn.